Among the following

a) pH of water decreases with increase in temperature

b) pH of water decreases by the addition of base

c) pH of water increases by the addition of acid

d) At any temperature pH of water is equal to $\frac{1}{2}p{K}_w$

The value of $K_w$ increases with rise in temperature

At high temperature dissociation of water is faster, so there will be more concentration of $\left[H_3{O}^{+}\right]$ ions & therefore pH decreases. On adding base pH is greater than 7 & on adding acid pH is less than 7.

$$K_w=\left[H_3{O}^{+}\right]\left[O{H}^{-}\right]$

Since$\left[H_3{O}^{+}\right]=\left[O{H}^{-}\right]$

$[H_3{O}^{+}{]}^2=K_w$

$\left[H_3{O}^{+}\right]=\sqrt{K_w}$

pH = $-log\left[H_3{O}^{+}\right]$

= $-log(K_w{)}^{\frac{1}{2}}$

Which is simplified to $\left(\frac{1}{2}\right)(-log(K_w)$, since $p{K}_w=-log{K}_w$

Therefore, for pure water pH = $\left(\frac{1}{2}\right)p{K}_w$

Hence, the correct option is option 3) a and d are correct.